* question about discrete op-fibrations @ 2012-02-01 0:03 David Spivak 2012-02-01 23:29 ` Mark Weber ` (2 more replies) 0 siblings, 3 replies; 5+ messages in thread From: David Spivak @ 2012-02-01 0:03 UTC (permalink / raw) To: categories list Hi all, Here's a quick question perhaps someone here can answer easily. Let DOF denote the category whose objects are small categories C,D, etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D. For a category C, let DOF_{C/} denote the coslice over C. Question: Does there exist a terminal object in DOF_{C/}? Thanks! David [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations 2012-02-01 0:03 question about discrete op-fibrations David Spivak @ 2012-02-01 23:29 ` Mark Weber [not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> 2012-02-02 10:22 ` Thorsten Palm 2 siblings, 0 replies; 5+ messages in thread From: Mark Weber @ 2012-02-01 23:29 UTC (permalink / raw) To: David Spivak; +Cc: categories list Dear David I'll assume by the coslice DOF_{C/} you mean the category whose objects are discrete opfibrations C --> D, and whose arrows are strictly commuting triangles under C. In that case the answer to your question is no. When C is empty, DOF_{C/} is just your category DOF, of small categories and discrete opfibrations between them, and DOF lacks a terminal object. For suppose that D is terminal in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(X) is the category obtained from X by freely adding an initial object. That is, the objects of I(X) are the elements of X together with one additional object 0, and one has a unique arrow 0 --> x for all x in X. If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such that the cardinality of the set of all arrows with source F(0) is that of X. Thus since D is terminal in DOF, for any set X there is an object x of D such that the cardinality of the set of all arrows with source x is that of X. This contradicts the smallness of D. With best regards, Mark Weber On 01/02/2012, at 11:03 AM, David Spivak <dspivak@gmail.com> wrote: > Hi all, > > Here's a quick question perhaps someone here can answer easily. > > Let DOF denote the category whose objects are small categories C,D, > etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D. > For a category C, let DOF_{C/} denote the coslice over C. > > Question: Does there exist a terminal object in DOF_{C/}? > > Thanks! > David > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
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* Re: question about discrete op-fibrations [not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> @ 2012-02-02 5:51 ` David Spivak 2012-02-03 14:07 ` Thorsten Palm 0 siblings, 1 reply; 5+ messages in thread From: David Spivak @ 2012-02-02 5:51 UTC (permalink / raw) To: Mark Weber; +Cc: categories list Hi Mark, Nice work; thank you for the simple answer and good explanation. I hope this isn't annoying, but what if I change the problem somewhat and take DOF(C) to be the full subcategory of Cat_{C/} spanned by the discrete opfibrations C-->D? Again I want to know whether DOF(C) has a terminal object. Under this definition, by setting C=empty-category we get DOF(C)=Cat, which does have a terminal object. Thanks, David On Wed, Feb 1, 2012 at 5:29 PM, Mark Weber <mark.weber.math@googlemail.com> wrote: > Dear David > > I'll assume by the coslice DOF_{C/} you mean the category whose objects are discrete opfibrations C --> D, and whose arrows are strictly commuting triangles under C. > > In that case the answer to your question is no. When C is empty, DOF_{C/} is just your category DOF, of small categories and discrete opfibrations between them, and DOF lacks a terminal object. For suppose that D is terminal in DOF. Then for any set X, there is a discrete opfibration I(X) --> D, where I(X) is the category obtained from X by freely adding an initial object. That is, the objects of I(X) are the elements of X together with one additional object 0, and one has a unique arrow 0 --> x for all x in X. > > If F:I(X) --> D is a discrete opfibration, then F(0) is an object of D such that the cardinality of the set of all arrows with source F(0) is that of X. Thus since D is terminal in DOF, for any set X there is an object x of D such that the cardinality of the set of all arrows with source x is that of X. This contradicts the smallness of D. > > With best regards, > > Mark Weber > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations 2012-02-02 5:51 ` David Spivak @ 2012-02-03 14:07 ` Thorsten Palm 0 siblings, 0 replies; 5+ messages in thread From: Thorsten Palm @ 2012-02-03 14:07 UTC (permalink / raw) To: David Spivak; +Cc: Mark Weber, categories list [ To all, moderator in particular, and with apologies to David: Please ignore my previous message. It contains a seriously wrong piece of information. ] Dear David, This time the answer is "yes" for discrete C, "no" otherwise. If C is discrete, the functor !_C : C-->1 is a discrete op-fibration, so that the terminal objet (1,!_C) of Cat_{C/} belongs to DOF(C). Now let C contain a non-trivial morphism u : c_0-->c_1 and suppose that (T,z) is terminal in DOF(C). Construct a category C_{c_1} by freely adding to C an object d and a morphism v : d-->c_1. Then we have an inclusion j : C-->C_{c_1}, which is a discrete op-fibration. We obtain two morphisms (C_{c_1},j)-->(T,z) in DOF(C) by applying z on the subcategory C and sending v to z(u) and the identity of z(c_1), respectively. By terminality of (T,z) they have to be the same. But since z is an op-fibration, z(u) cannot be an identity --- contradiction. (I hope Mark hasn't beaten me to it again.) Thorsten David Spivak hat am 01.02.12 geschrieben: > > I hope this isn't annoying, but what if I change the problem somewhat > and take DOF(C) to be the full subcategory of Cat_{C/} spanned by the > discrete opfibrations C-->D? Again I want to know whether DOF(C) has a > terminal object. Under this definition, by setting C=empty-category we > get DOF(C)=Cat, which does have a terminal object. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
* Re: question about discrete op-fibrations 2012-02-01 0:03 question about discrete op-fibrations David Spivak 2012-02-01 23:29 ` Mark Weber [not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> @ 2012-02-02 10:22 ` Thorsten Palm 2 siblings, 0 replies; 5+ messages in thread From: Thorsten Palm @ 2012-02-02 10:22 UTC (permalink / raw) To: David Spivak; +Cc: categories list David Spivak hat am 31.01.12 geschrieben: > > Let DOF denote the category whose objects are small categories C,D, > etc. and in which Hom(C,D) is the set of discrete op-fibrations C-->D. > For a category C, let DOF_{C/} denote the coslice over C. > > Question: Does there exist a terminal object in DOF_{C/}? Answer: No. The "op-" is irrelevant; let us look at DF_{C/} instead. There is not even a weakly terminal object, for the same reason for which there isn't one in DF itself. (Suppose (T,z) is one. Let A be an arbitrary small category having a terminal object a. From the object (C+A,incl) of DF_{C/} we get a discrete fibration f : A-->T. But then for t = f(a) the slice category T_{/t} is isomorphic to A. There are not enough t to accommodate all possible A.) Thorsten Palm [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 5+ messages in thread
end of thread, other threads:[~2012-02-03 14:07 UTC | newest] Thread overview: 5+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2012-02-01 0:03 question about discrete op-fibrations David Spivak 2012-02-01 23:29 ` Mark Weber [not found] ` <81982979-2217-4AC4-AEDD-154DB2EEAC7B@gmail.com> 2012-02-02 5:51 ` David Spivak 2012-02-03 14:07 ` Thorsten Palm 2012-02-02 10:22 ` Thorsten Palm
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